The origin of the large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\mathrm{c}}$\end{document}Tc variation in FeSe thin films probed by dual-beam pulsed laser deposition

FeSe is one of the most enigmatic superconductors. Among the family of iron-based compounds, it has the simplest chemical makeup and structure, and yet it displays superconducting transition temperature (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\text{c}}$\end{document}Tc) spanning 0 to 15 K for thin films, while it is typically 8 K for single crystals. This large variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\text{c}}$\end{document}Tc within one family underscores a key challenge associated with understanding superconductivity in iron chalcogenides. Here, using a dual-beam pulsed laser deposition (PLD) approach, we have fabricated a unique lattice-constant gradient thin film of FeSe which has revealed a clear relationship between the atomic structure and the superconducting transition temperature for the first time. The dual-beam PLD that generates laser fluence gradient inside the plasma plume has resulted in a continuous variation in distribution of edge dislocations within a single film, and a precise correlation between the lattice constant and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\text{c}}$\end{document}Tc has been observed here, namely, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\text{c}} \propto \sqrt{c- c_{0}}$\end{document}Tc∝c−c0, where c is the c-axis lattice constant (and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{0}$\end{document}c0 is a constant). This explicit relation in conjunction with a theoretical investigation indicates that it is the shifting of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{\text{xy}}$\end{document}dxy orbital of Fe which plays a governing role in the interplay between nematicity and superconductivity in FeSe. Supplementary Information The online version contains supplementary material available at 10.1007/s44214-024-00058-0.

result in distorted (020) planes.While these distortions are a few nanometers in size, the typical thickness of our STEM specimen is 30 -50 nm, therefore the projection we observe always include contributions from both distorted and undistorted planes.Panels (d) and (e) schematically illustrate that the distorted region is the overlap between a distorted plane and an undistorted plane.The in-plane lattice of the distorted plane is stretched compared to that of the standard plane.Such in-plane expansion introduces stress in the entire sample, leading to the increase of in-plane lattice constant and decrease of out-of-plane lattice constant.
(  ) MD: Here,   represents the laser source,   is the electron specific heat,   is the lattice specific heat,  is lattice-electron coupling factor,   (  l ) is the electron (lattice) thermal conductivity described by the Drude model.In the MD system,   is the number of cells and the Newton's equations of motion of atoms have the thermal velocity friction term with  the friction constant for the energy dissipation.In the simulation, the processes of melting as well as the ablation are confined within the MD region of the model.
In the second step, we assume that the plume will evolve from a non-equilibrium state to an equilibrium state across the thin Knudsen layer.The plume expansion is described by the fluid dynamics with the governing equations: where   denotes the density of ambient gas,  is the plume density,  → is the velocity of the flume,  evap is the total pressure of the ambient gas and the plume vapor.In order to obtain the boundary condition, the spatial and velocity distributions of atoms calculated by the MD-TTM simulation within several hundred-nanometer scales are mapped onto the Knudsen layer of the fluid model.In this mapping, the thermal evaporation model ( 2) is used , in which the vapor pressure at the surface is obtained from the Clasius-Clapeyron equation.
We also follow the conservation of mass, momentum, and energy across the Knudsen layer when applying the boundary conditions for the fluid model.
In addition, we have performed first-principles dynamics simulations of bulk FeSe under intense laser irradiation based on real-time time-dependent density functional theory.
4).The Vienna ab initio software package ( 5) is employed to perform the DFT calculations (6).The projector augmented-wave method is used to describe the wave functions near the core.The generalized gradient approximation (7) within the Perdew-Burke-Ernzerhof (PBE) (8) parameterization is employed as the electron exchange-correlation function.The primitive cell of FeSe (space group P4/nmm) is used for calculations with the Brillouin zone sampling of 6×6×4.DFT-D2 treatment is employed to correct for the van der Waals interactions in layered FeSe (9).All magnetic ions are initialized ferromagnetically, and the cell shape, volume, and atomic positions are fully optimized throughout this work.After the structural optimization, the final structure of the slightly electron-doped FeSe shows that there is no spin polarization when the doping level is low.A small amount of excess Fe in the composition will lead to electron doping in the compounds.Thus, a 0.5 % increase in Fe concentration in composition can be associated with a 0.05 Å lattice expansion in c direction.
As For band structure calculations, the cutoff energy of 500 eV is used for expanding the wave functions into plane-wave basis.In the calculation, the Brillouin zone is sampled in the k space within Monkhorst-Pack scheme (10).The number of the k points is 11×11×7.We relax internal atomic positions, and forces are minimized to less than 0.01 eV/Å in the relaxation.

Figure S3 |
Figure S3 | θ-2θ diffraction patterns.(a) FeSe film made by the conventional single-laser PLD method.(b) FeSe film made by the dual-beam PLD method."FS" and "CF" stand for FeSe and CaF 2 (substrate), respectively.Different colors in panel (b) represent data taken at different locations of the film using an x-ray beam with a width of ≈ 0.4 mm.Detailed evolution of the peak as a function of the position is shown in Fig. 1(f) of the main text.

Figure S4 |
Figure S4 | Evolution of the in-plane lattice parameter of the dual-beam PLD FeSe film across the substrate.(a) The XRD patterns of the (220) peak along the x-direction (as denoted in Fig. 1(e) of the main text) from -15 to +15 mm.The grazing-incident x-ray beam is aligned parallel to the ab-plane of the FeSe film.A clear continuous shift is observed, first to higher angle approaching the middle and then the shifting is reversed in a symmetrical manner.(b) The correlation between the a-axis lattice constant (extracted from panel (a)) and T c .

Figure S5 |
Figure S5 | Comparison of the laser energy distribution for the single-and dual-beam PLD.(a-b) Single-(a) and dual-(b) beam PLD configurations.(c-d) The laser energy distribution for the single-(c) and dual-(d) beam configurations.

Figure S6 |
Figure S6 | Variation of T c and lattice parameters across the dual-beam FeSe film.From bottom to the top, panels show the position (x-direction) dependence (across the substrate) of T c , c-axis, a-axis and unit cell volume (V).

Figure S7 |
Figure S7 | Schematic illustration of the distorted region and its relation with the variation of the lattice constants.Panel (a) is a magnified image of the area enclosed by the red box in panel (b), which is the HAADF image of the sample with T c of 3 K along the [100] projection.Panel (c) is the inverse fast Fourier transform (IFFT) image with (020) spots, and panel (f) is the blow-up image of the area enclosed by the red box in panel (c).Panels (c) and (f) show there exist partial edge dislocations (i.e. an extra half-unit-cell plane inserted between two nearby planes, highlighted by the green circle and arrows in panel (c)), which

Figure S8 |
Figure S8 | Statistics of the distorted areas observed in a low-T c FeSe film deposited by conventional single-beam PLD.The boxes roughly capture the distorted regions in each image.Overall, the distorted regions occupy around 10% of the total volume.

Figure S9 |
Figure S9 | Thickness determination for a dual-beam FeSe film.(a-b) The scanning electron microscope (SEM) images for the dual-beam FeSe film at the center (a) and edge (b) of the substrate, separated by ≈ 15 mm.The thicknesses (labeled in the figures) are 201 (± 5) and 203 (± 5) nm for panels (a) and (b), respectively, showing little variation from the center to the edge.This result confirms that the thickness of the dual-beam FeSe film is uniform across the substrate.

Figure S11 |
Figure S11 | Temperature dependence of the normalized resistance of FeSe films made by single-beam PLD.R N corresponds to the resistance at 20 K.The samples show very sharp superconducting transitions (e.g.ΔT ≈ 0.3 K for the SC12 sample), suggesting the high qualities of the FeSe films.

Figure S12 |
Figure S12 | The normal-state Hall resistivity ρ xy (B) for five FeSe films (SC14, SC12, SC11, SC08 and SC03) made by the single-beam PLD.The Hall resistivity is proportional to the magnetic field up to 9 T at all measured temperatures from 200 to 20 K, indicating that the Hall coefficient is field independent.

Figure. S13 |
Figure.S13 | Electronic band structure of FeSe with varying lattice parameters.Band structures around Γ and M points calculated from our experimental crystal lattice parameter data taken at room temperature.Here, we show band structures based on three sets of lattice constants (Supplementary Note 5): a purple > a red > a orange and c purple < c red < c orange .The d xy orbitals are below the E F around Γ, but pass through the E F around M. With the lattice modulation, the most noticeable change in the electronic structures takes place in the d xy band: the d xy band shifts up in energy around Γ and shifts down around M, while d xz /d yz bands exhibit little change.
mentioned in the main text, such an electron-filling picture can neither explain how the value of the Hall coefficients for different samples here overlap with each other above T * (as seen in Fig 4(e)), nor be the reason for the hole carrier concentration remaining almost constant.Thus, we rule out the composition change as the cause of the observed lattice constant change and the concomitant T c variation across the FeSe film deposited by the dual-beam PLD.
The adopted lattice constants are: a = 3.773 Å, c = 5.505 Å for purple lines, a = 3.763 Å, c = 5.530 Å for red lines and a =3.733Å, c = 5.562 Å for orange lines in Fig. S13.The key findings are as follows: with decreasing a-axis parameter and increasing c-axis parameter, the most noticeable change in the electronic structures takes place in the d xy band: the d xy band shifts up in energy around Γ and shifts down around M, while d xz /d yz bands exhibit little change (Fig. S13).Phenomenologically, as the diagonal hopping between d xy orbitals is sensitive to Se height while d xz /d yz is not, a small variation in Se height will mainly affect the d xy bands.With a decrease in a-axis, the Se height increases and this leads to a reduction of the coupling between Fe d xy and Se p x /p y orbitals.Because the effective hopping between d xy orbitals from the above coupling is negative, this reduction is expected to increase the diagonal hopping between d xy orbitals, which, in turn, causes the d xy band's upshift around Γ and downshift around M. It should be noted that all DFT calculations are done for the tetragonal phase (at high temperatures, i.e.T > T*), where the d xy electron Fermi pocket shows only slight changes, consistent with the overlapped Hall coefficient at high temperatures.The experimentally-observed dramatic change of the electron density occurs at low temperatures.Due to the fact that the nematic phase and other strong correlation effects at low temperatures cannot be fully captured in DFT, the absolute number of the change in the electron density cannot be extracted from our calculations.Nevertheless, the most important piece of information from the calculations here is that the band dispersion of the d xy orbital is more sensitive to the lattice change than the d xz / yz bands.The shift of the d xy orbital bands will have a significant effect on the corresponding band splitting at low temperatures (nematic phase), which may lead to a more dramatic change in the electron density.